---
title: Ryan–Smith Localization in Set Theory
url: https://www.emergentmind.com/topics/ryan-smith-localization
type: topic
---

# Ryan–Smith Localization in Set Theory

Ryan–Smith localization is a framework in set theory that establishes equivalences between global and local forms of the Partition Principle (PP) and restricted variants of the Axiom of Choice, under certain "small-choice" hypotheses. It underpins constructive separations of $\mathsf{PP}$ from $\mathsf{AC}$ in symmetric models, solving longstanding open questions in the theory of choice principles.

## 1. Fundamental Definitions

The Partition Principle (PP) states: for every surjection $f: A \twoheadrightarrow B$, there exists an injection $i: B \hookrightarrow A$. In terms of cardinal arithmetic, this asserts $|B| \leq |A|$ whenever $|B| \leq^* |A|$. Local restriction of PP to a set $T$ (written $\mathsf{PP} \restriction T$ or $\mathsf{PP} \!\restriction T$) means the surjection-to-injection property is asserted only for surjections where both domain and codomain are subsets of $T$. The Axiom of Choice restricted to well-ordered index sets, denoted $\mathsf{AC}_{\mathsf{WO}}$, asserts that every surjection $g: Y \twoheadrightarrow \lambda$ (with $\lambda$ an ordinal) admits a right inverse $s: \lambda \to Y$, equivalently, every well-ordered family of nonempty sets has a choice function.

"Small Violations of Choice" (SVC) for a parameter set $S$ holds when every set $X$ is a surjective image of $S \times \eta$ for some ordinal $\eta$; formally, $\mathrm{SVC}(S) \equiv \forall X \ \exists \eta \ \exists f: S \times \eta \twoheadrightarrow X$. The strengthening $\mathrm{SVC}^+(S)$ replaces "surjective image" with "injective image": $\forall X \ \exists \eta \ \exists i: X \hookrightarrow S \times \eta$. A standard consequence due to Ryan–Smith is that $\mathrm{SVC}(S) \implies \mathrm{SVC}^+(\mathcal{P}(S))$ when $T = \mathcal{P}(S)$, with $S$ a set of sequences of Cohen reals in the cited constructions [2601.01855].

## 2. The Ryan–Smith Localization Theorem

The Ryan–Smith Localization Theorem asserts:  
Assume $\mathrm{SVC}^+(T)$. In ZF,
$$
\mathsf{PP} \ \Longleftrightarrow \ [\mathsf{PP} \!\restriction T \ \wedge\ \mathsf{AC}_{\mathsf{WO}}].
$$
That is, under the small-choice hypothesis for parameter $T$, the global Partition Principle is equivalent to its local restriction to $T$ together with the Axiom of Choice for well-ordered families. This equivalence is a central technical tool for separating choice principles at the level of symmetric extensions and iterated forcing.

A summary of key relationships is presented in the following table:

| Principle             | Domain                      | Content                                      |
|-----------------------|----------------------------|----------------------------------------------|
| $\mathsf{PP}$         | All sets                    | Surjections split via injections             |
| $\mathsf{PP}\!\restriction T$ | $X,Y \subseteq T$        | As above, restricted to $T$                  |
| $\mathsf{AC}_{\mathsf{WO}}$   | Ordinals/indexed families | Choice on well-ordered families              |
| $\mathrm{SVC}^+(T)$   | All sets                    | Each set injects into $T\times\eta$ for some $\eta$ |

## 3. Outline of the Equivalence Proof

The proof, as presented in [2601.01855] and originally established in Proposition 3.17 of C. Ryan-Smith ("Local reflections of choice", Acta Math. Hung. 2025), proceeds as follows:

- **$(\Rightarrow)$ Direction:** Global $\mathsf{PP}$ trivially entails the local restriction $\mathsf{PP}\!\restriction T$. The assertion $\mathsf{AC}_{\mathsf{WO}}$ follows since any surjection $Y \twoheadrightarrow \lambda$ (with $\lambda$ an ordinal) is a special case of $\mathsf{PP}$.
- **$(\Leftarrow)$ Direction:** Assume $\mathrm{SVC}^+(T)$, $\mathsf{PP}\!\restriction T$, and $\mathsf{AC}_{\mathsf{WO}}$. Let $f: Y \twoheadrightarrow X$ be arbitrary. By $\mathrm{SVC}^+(T)$, there exist $\eta$ and $j: X \hookrightarrow T \times \eta$. For each $\alpha < \eta$, let $X_\alpha = \{ t \in T : \exists x(j(x) = (t,\alpha)) \}$, $Y_\alpha = f^{-1}[X_\alpha]$, giving surjections $f_\alpha: Y_\alpha \twoheadrightarrow X_\alpha$. By $\mathsf{PP}\!\restriction T$, each $f_\alpha$ splits via $s_\alpha: X_\alpha \rightarrow Y_\alpha$. As $\eta$ is well-ordered, $\mathsf{AC}_{\mathsf{WO}}$ assembles the $s_\alpha$ into a global injection $s: X \rightarrow Y$ with $f \circ s = \mathrm{id}_X$, establishing $\mathsf{PP}$.

## 4. Applications in Symmetric Models and Independence Results

Ryan–Smith localization facilitates independence proofs regarding the relative strength of set-theoretic choice principles. In [2601.01855], the theorem is leveraged to construct a transitive model $M$ satisfying $\mathrm{ZF} + \mathrm{DC} + \mathsf{PP} + \neg \mathsf{AC}$ via class-length countable-support symmetric iterations from a Cohen symmetric seed. In such models:
- $\mathrm{SVC}(S)$ holds for $S = A^\omega$ (where $A$ is the Cohen-reals set),
- $\mathsf{PP} \!\restriction T$ and $\mathsf{AC}_{\mathsf{WO}}$ hold for $T = \mathcal{P}(S)$,
- Yet $\neg \mathsf{AC}$ is forced, as $A$ is not well-orderable.

The implication $\mathrm{SVC}(S) \implies \mathrm{SVC}^+(T)$ and the localization theorem then yield $M \models \mathsf{PP}$. Therefore, these constructions formally establish that $\mathsf{PP}$ does not imply $\mathsf{AC}$.

## 5. Infrastructure for Symmetric Iterations and Preservation Properties

The construction of relevant symmetric models involves:
- Countable-support symmetric iterations,
- Successor stages forcing with "orbit-symmetrized packages" $Q_{[f]}$ (to split surjections within $T$) and $R_{[g]}$ (to split surjections onto ordinals $< \aleph^*(S)$),
- Diagonal-cancellation/diagonal-lift infrastructures supplying $\omega_1$-complete normal filters at limit stages,
- Ensuring generic sections are hereditarily symmetric and preservation of $\mathrm{DC}$ via normality and $\omega_1$-completeness.

This infrastructure is critical for maintaining fine control over the properties of the constructed model at each stage and for preserving the delicate balance required to force $\mathrm{PP}$ without $\mathrm{AC}$.

## 6. Historical and Technical Significance

Ryan–Smith localization addresses a problem posed by Russell in 1906 concerning the equivalence of $\mathsf{PP}$ and $\mathsf{AC}$ and clarifies which fragments of choice are necessary for $\mathsf{PP}$ in certain models. The framework enables local-to-global reductions, crucial for advancing set theory without choice and constructing intermediate models between $\mathsf{ZF}$ and $\mathsf{ZF} + \mathsf{AC}$. The results are summarized and expanded upon in Gilson [2601.01855], §4.2 ("Reduction blueprint") and Theorem 4.4, with full details in Ryan-Smith ("Local reflections of choice", Acta Math. Hung. 2025).

Source: https://www.emergentmind.com/topics/ryan-smith-localization